Orthogonal Digital Computation in Two-Digit Squares: A Carry-Free Approach for Simultaneous Leader and Digit Processing

Title: Orthogonal Digital Computation in Two-Digit Squares: A Carry-Free Approach for Simultaneous Leader and Digit Processing Abstract: This paper introduces the Orthogonal Digital Computation (ODC) algorithm, a methodological innovation in mental and digital arithmetic that eliminates sequential carry propagation for squaring two-digit numbers (1–99). Unlike traditional methods that rely on interdependent digit steps, ODC decomposes the squaring process into three autonomous digital blocks—Units, Tens, and Leaders (hundreds/thousands)—processed in complete computational orthogonality. The theoretical framework is anchored in the Maltese Cross, a 14x14 symmetric matrix that implements a truncated binary partition of the number set based on modular symmetries. By applying the Three C Rule (Constant, Complement, Coincident), the algorithm identifies symmetry centers to determine each result digit independently without inter-block dependencies. This approach significantly reduces cognitive load by eliminating the need to maintain intermediate carries in working memory, offering a fundamental departure from conventional sequential architectures